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 AM  Vol.7 No.10 , June 2016
Multigrid Method for the Numerical Solution of the Modified Equal Width Wave Equation
Abstract: Numerical solutions of the modified equal width wave equation are obtained by using the multigrid method and finite difference method. The motion of a single solitary wave, interaction of two solitary waves and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Using error norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other methods.
Cite this paper: Essa, Y. (2016) Multigrid Method for the Numerical Solution of the Modified Equal Width Wave Equation. Applied Mathematics, 7, 1140-1147. doi: 10.4236/am.2016.710102.
References

[1]   Brandt, A. (1977) Multi-Level Adaptive Solution to Boundary-Value Problem. Mathematics of Computation, 31, 333- 390.
http://dx.doi.org/10.1090/S0025-5718-1977-0431719-X

[2]   Abo Essa, Y.M., Amer, T.S. and Ibrahim, I.A. (2011) Numerical Treatment of the Generalized Regularized Long- Wave Equation. Far East Journal of Applied Mathematics, 52, 147-154.
https://www.researchgate.net/.../266611672

[3]   Abo Essa, Y.M., Amer, T.S. and Abdul-Moniem, I.B. (2012) Application of Multigrid Technique for the Numerical Solution of the Non-Linear Dispersive Waves Equations. International Journal of Mathematical Archive, 3, 4903- 4910.

[4]   Abo Essa, Y.M., Ibrahim, I.A. and Rahmo, E.-D. (2014) The Numerical Solution of the MRLW Equation Using the Multigrid Method. Applied Mathematics, 5, 3328-3334.
http://dx.doi.org/10.4236/am.2014.521310

[5]   Morrison, P.J., Meiss, J.D. and Cary, J.R. (1984) Scattering of Regularized-Long-Wave Solitary Waves. Physica D. Nonlinear Phenomena, 11, 324-336.
http://dx.doi.org/10.1016/0167-2789(84)90014-9

[6]   Gardner, L.R.T. and Gardner, G.A. (1990) Solitary Waves of the Regularized Long Wave Equation. Journal of Computational Physics, 91, 441-459.

[7]   Gardner, L.R.T. and Gardner, G.A. (1992) Solitary Waves of the Equal Width Wave Equation. Journal of Computational Physics, 101, 218-223.
http://dx.doi.org/10.1016/0021-9991(92)90054-3

[8]   Abdulloev, Kh. O., Bogolubsky, I.L. and Makhankov, V.G. (1976) One More Example of Inelastic Soliton Interaction. Physics Letters. A, 56, 427-428.
http://dx.doi.org/10.1016/0375-9601(76)90714-3

[9]   Gardner, L.R.T., Gardner, G.A. and Geyikli, T. (1994) The Boundary Forced MKdV Equation. Journal of Computational Physics, 113, 5-12.

[10]   Geyikli, T. and Battal Gazi Karakoc, S. (2011) Septic B-Spline Collocation Method for the Numerical Solution of the Modified Equal Width Wave Equation. Applied Mathematics, 2, 739-749.
http://dx.doi.org/10.4236/am.2011.26098

[11]   Geyikli, T. and Battal Gazi Karakoc, S. (2012) Petrov-Galerkin Method with Cubic B-Splines for Solving the MEW Equation. Bulletin of the Belgian Mathematical Society - Simon Stevin, 19, 215-227.
http://projecteuclid.org/euclid.bbms/1337864268

[12]   Esen, A. (2005) A Numerical Solution of the Equal width Wave Equation by a Lumped Galerkin Method. Applied Mathematics and Computation, 168, 270-282.
http://dx.doi.org/10.1016/j.amc.2004.08.013

[13]   Esen, A. (2006) A Lumped Galerkin Method for the Numerical Solution of the Modified Equal-Width Wave Equation Using Quadratic B-Splines. International Journal of Computer Mathematics, 83, 449-459.
http://dx.doi.org/10.1080/00207160600909918

[14]   Saka, B. (2007) Algorithms for Numerical Solution of the Modified Equal Width Wave Equation Using Collocation Method. Mathematical and Computer Modeling, 45, 1096-1117.
http://dx.doi.org/10.1016/j.mcm.2006.09.012

[15]   Zaki, S.I. (2000) Solitary Wave Interactions for the Modified Equal Width Equation. Computer Physics Communications, 126, 219-231.
http://dx.doi.org/10.1016/S0010-4655(99)00471-3

[16]   Zaki, S.I. (2000) Least-Squares Finite Element Scheme for the EW Equation. Computer Methods in Applied Mechanics and Engineering, 189, 587-594.
http://dx.doi.org/10.1016/S0045-7825(99)00312-6

[17]   Wazwaz, A.-M. (2006) Thetanh and the Sine-Cosine Methods for a Reliable Treatment of the Modified Equal Width Equation and Its Variants. Communications in Nonlinear Science and Numerical Simulation, 11, 148-160.
http://dx.doi.org/10.1016/j.cnsns.2004.07.001

[18]   Saka, B. and Da?, ?. (2007) Quartic B-Spline Collocation Method to the Numerical Solutions of the Burgers’ Equation, Chaos. Solitons and Fractals, 32, 1125-1137.
http://dx.doi.org/10.1016/j.chaos.2005.11.037

[19]   Lu, J. (2009) He’s Variational Iteration Method for the Modified Equal Width Equation. Chaos, Solitons and Fractals, 39, 2102-2109.
http://dx.doi.org/10.1016/j.chaos.2007.06.104

[20]   Evans, D.J. and Raslan, K.R. (2005) Solitary Waves for the Generalized Equal Width (GEW) Equation. International Journal of Computer Mathematics, 82, 445-455.
http://dx.doi.org/10.1080/0020716042000272539

[21]   Esen, A. and Kutluay, S. (2008) Solitary Wave Solutions of the Modified Equal Width Wave Equation. Communications in Nonlinear Science and Numerical Simulation, 13, 1538-1546.
http://dx.doi.org/10.1016/j.cnsns.2006.09.018

[22]   Battal Gazi Karakoc, S. and Geyikli, T. (2012) Numerical Solution of the Modified Equal width Wave Equation. International Journal of Differential Equations, 2012, Article ID: 587208.
http://dx.doi.org/10.1155/2012/587208

 
 
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